How to make assemblies from multiple objects?
Let's start with the hole object and the shaft object (shown below). I've taken the opportunity to rotate the hole from the global XYZ axes just in case that's what you're dealing with.



The first thing you want to do is to add an empty axis along the desired rotational axis of the shaft. If the shaft is symmetrical and aligned with the global axes, select the shaft, use Shift+S > Cursor to Selected to move the 3D Cursor to the shaft origin, and add Shift+A > Empty > Plain Axis. You'll then want to parent the shaft to the Plain Axis.

That step is to allow you to rotate the shaft about its local X, Y or Z-axis. Long story short, if you try to spin an object around an axis that's not the global axis, things get messy. In the case of your shaft not being aligned to any of the XYZ axes, you'll have to first rotate the shaft so that the rotational axis does align with one of them, then press Ctrl+A > Apply Rotation. Then you can add the empty as I showed above.



Next, go to the hole. Select all the vertices/edges that make up the hole, press F to fill it in, then press Ctrl+F > Poke Faces. What this step will do is create a face with a normal that matches the desired rotational axis, and adds a vertex in the middle of it that we can use to snap the Plain Axis to.



Next, enable Vertex Snapping making sure you enable Align Rotation to Target. Grab the Plain Axis, and snap it to the vertex we made in the hole earlier.




Now you can delete the vertex from the hole, and rotate the shaft by rotating the shaft object, leave the Plain Axis alone.



EDIT: from additional info

If you don't need to rotate the shaft, you don't need to use the Plain Axis. Fill in the hole face, and poke it as I showed above. Then, if you grab the shaft using Vertex Snapping, Snap with Active Element and Align Rotation to Target, you can move it onto the hole and it will automatically align itself for you. Then you can delete the extra vertex from poking the face.

From there, if you need the shaft to be further in/out of the whole, use G+Z+Z to slide it in/out.